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In our recent paper [H. Baran, I.S. Krasil'shchik, O.I. Morozov, P.Voj{\v{c}}{\'{a}}k, Symmetry reductions and exact solutions of Lax integrable$3$-dimensional systems, Journal of Nonlinear Mathematical Physics, Vol. 21,No. 4 (December 2014), 643--671; arXiv:1407.0246 [nlin.SI], DOI:10.1080/14029251.2014.975532}], we gave a complete description of symmetryreduction of four Lax-integrable (i.e., possessing a zero-curvaturerepresentation with a non-removable parameter) $3$-dimensional equations. Herewe study the behavior of the integrability features of the initial equationsunder the reduction procedure. We show that the ZCRs are transformed tononlinear differential coverings of the resulting 2D-systems similar to the onefound for the Gibbons-Tsarev equation in [A.V. Odesskii, V.V. Sokolov,Non-homogeneous systems of hydrodynamic type possessing Lax representations,arXiv:1206.5230, 2006]. Using these coverings we construct infinite series of(nonlocal) conservation laws and prove their nontriviality. We also show thatthe recursion operators are not preserved under reductions.

Integrability properties of some symmetry reductions